.experimental::FreeOperad – domain for the free operad
experimental::FreeOperad(R) creates a domain for the free operad on one binary generator over the ground ring .
experimental::FreeOperad(R, [f,g,], ...): similar-wise with several binary generators.
experimental::FreeOperad(R, [f=[Arity=3, Commutative],g=[Arity=1],], ...): similar-wise, but specifying further properties of the generators (to be implemented).
We create the free operad over two binary operators f and g:
A := experimental::FreeOperad(Dom::Rational, [f, g]):
The basis of this operad consists of all the possible compositions of f and g; here are those compositions which are ternary:
A::basis::list(3)
-- f , f , f , f , f , f , f , f , f , f , f , f , f , f , f , f , f , f , f ,
| / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \
| 1 f 1 f 1 g 1 g 2 f 2 f 2 g 2 g 3 f 3 f 3 g 3 g f 3 f 3 g 3 g 3 f 2 f 2 g 2
| / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \
-- 2 3 3 2 2 3 3 2 1 3 3 1 1 3 3 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 3 3 1 1 3
f , f , f , f , f , g , g , g , g , g , g , g , g , g , g , g , g , g , g ,
/ \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \
g 2 f 1 f 1 g 1 g 1 1 f 1 f 1 g 1 g 2 f 2 f 2 g 2 g 3 f 3 f 3 g 3 g f 3 f 3
/ \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \
3 1 2 3 3 2 2 3 3 2 2 3 3 2 2 3 3 2 1 3 3 1 1 3 3 1 1 2 2 1 1 2 2 1 1 2 2 1
g , g , g , g , g , g , g , g , g , g --
/ \ / \ / \ / \ / \ / \ / \ / \ / \ / \ |
g 3 g 3 f 2 f 2 g 2 g 2 f 1 f 1 g 1 g 1 |
/ \ / \ / \ / \ / \ / \ / \ / \ / \ / \ |
1 2 2 1 1 3 3 1 1 3 3 1 2 3 3 2 2 3 3 2 --
Changes in MuPAD 3.1
New Function.